www.mathcentre.ac.uk/.../Composition%20of%20functions.mp4

The composition of two functions
G&F is the function we get from

doing at 1st and then G. So if
we let F of X.

Be given by F of X
equals X squared.

And we let G of X.

BX +3.

Then GF of X is given by.
Well, we do F of X first and

that's X squared.

Then we apply G to the
whole of that.

Now, since G just adds 3 onto a

number. G of X squared is
X squared +3.

So GF of X is X squared +3.

Here's another example.

This time will let F of X.

Be given by F of X equals 2X.

And we'll have G of X.

Being East The X.

Now to get GFX again, we write
down what F of X is.

That was 2X.

And then we apply G
to the whole of that.

Now G of X is E to the X.

So G of two X is E to the 2X.

So again we have here GF of X is
E to the 2X.

Sometimes the composition of two
functions is called the function

of a function, and here's
another way of writing it down.

We sometimes writes GF of
X as G Little Circle F

of X.

Now you don't have to use this
notation yourself, but you

might see it in classes or
workbooks, so it's important

to know what it means.

The order in which we compose
functions can make a big

difference. If you remember our
first example, we had F of X

being X squared.

And G of X.

Being X +3.

We'll see what happens when we
do G 1st and then F, so this is

FG of X.

So this time we write down what
G of X is first, which is X +3.

Then we apply F to
the whole of that.

Now applying the F to something
just makes it square, so F of X

+3 is X plus three all squared.

And that X squared plus
6X plus 9.

And if you remember, GF of X was
just X squared +3.

So you can see here that
changing the order in which

we compose these functions
makes a big difference.

Now we've looked at how to
compose functions. We look at

how to decompose functions.

If I have a function like each
of X equals E to the 2X.

We've already seen that we
can write H as a composition

of two functions.

H of X was equal to GF
of X, where F of X was

2X and G of X was E
to the X.

Let's decompose another
function. This time will have

function each of X.

Equals the cube roots.

Of 2X plus one.

The first here we send X to 2X
Plus One and then we cube root

the whole thing. So if we let F
of XB2X plus one then H of X.

Is the cube root?

Oh, this is X.

So then if we sat G of X.

To be cube root X.

We get that HX.

Is G.

Of F of X. So that's
GF of X.

Being able to decompose
functions will be very

important when you come to do
things like integration and

Differentiation.

Now there is some pairs of
functions that can't be

composed. Let's have F of
X being minus X squared.

And we'll have G of X.

Being natural log of X.

Now F of X.

Is less than or equal to 0?

Whichever X you pick.

But G of X is only defined if X
is greater than 0.

So G of F of X.

Doesn't exist.

There are also some pairs of
functions that can be

composed with some ex, but
not for other ex.

This time will have F of
X equals 4X minus 6.

And we'll have geovax.

Bing square root of X.

Now G of X is only defined if X
is greater than or equal to 0.

So GF of X can only be
defined if F of X is greater

than or equal to 0.

And F of X is greater than or
equal to 0.

Only if X is greater than or
equal to three over 2.

So GF of X is only
defined.

4X greater than or equal to
three over 2.

The values of X for which
GF is defined is called the

domain of GF.

The range of a function is the
set of all the values of

function can take.
So for example, if F

of X was each the X.

The range.

Is given by F of X is greater
than zero because E to the X can

only be greater than 0.

Similarly, if we have F of X
being say, Sign X.

Then the range.

Is FX.

Is between plus one and
minus one.

The range of a composite
function must lie within the

range of the second function we

apply. Here's an example to show

you this. Will put
down F of X equals X

minus 8 and G of X?

Equals X squared.

Now for F of X.

The range.

Is well F of X can take
any value.

But for G of X.

Ranges G of X is greater than or
equal to 0.

Now let's look at the range of
the composed function GF.

GFX is well F of X
is X minus 8.

And G of X squares it, so
GF of X is X minus 8

squared. Which is X
squared minus 16X plus

64.

But since GF of X is
something squared.

And all square numbers are
greater than or equal to 0.

GF of X is greater
than or equal to 0.

So the range here.

Is GF of X is greater than or
equal to 0?

So you can see here that the
range of GF is the range of G.

Now let's look at what
happens when he composed

and the other way around.

So this time FG books.

What again G of X is X squared?

And applying eh? The whole of
that gives us X squared minus 8.

Now, since G of X is greater
than or equal to 0.

And here we're taking away 8.

This whole thing must be greater
than or equal to minus 8.

So the range.

Is FGX is greater than or equal
to minus 8?

Now FG of X is range isn't the
same as F, of X is range.

But it certainly contained in
the range of F of X.

So these two examples show you
that if you're given a composed

function, the range must lie
within the range of the second

function you apply, but doesn't
have to be all of it.